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Theorem archiabl 30755
Description: Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
Assertion
Ref Expression
archiabl ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Proof of Theorem archiabl
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2821 . . . . 5 (0g𝑊) = (0g𝑊)
3 eqid 2821 . . . . 5 (le‘𝑊) = (le‘𝑊)
4 eqid 2821 . . . . 5 (lt‘𝑊) = (lt‘𝑊)
5 eqid 2821 . . . . 5 (.g𝑊) = (.g𝑊)
6 simpll1 1204 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
7 simpll3 1206 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
8 simplr 765 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑣 ∈ (Base‘𝑊))
9 simprl 767 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → (0g𝑊)(lt‘𝑊)𝑣)
10 simp2 1129 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑦 ∈ (Base‘𝑊))
11 simp1rr 1231 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
12 simp3 1130 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → (0g𝑊)(lt‘𝑊)𝑦)
13 breq2 5062 . . . . . . . 8 (𝑥 = 𝑦 → ((0g𝑊)(lt‘𝑊)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑦))
14 breq2 5062 . . . . . . . 8 (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥𝑣(le‘𝑊)𝑦))
1513, 14imbi12d 346 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1615rspcv 3617 . . . . . 6 (𝑦 ∈ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) → ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1710, 11, 12, 16syl3c 66 . . . . 5 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑣(le‘𝑊)𝑦)
181, 2, 3, 4, 5, 6, 7, 8, 9, 17archiabllem1 30750 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
1918adantllr 715 . . 3 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
20 simpr 485 . . . 4 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
21 breq2 5062 . . . . . 6 (𝑢 = 𝑣 → ((0g𝑊)(lt‘𝑊)𝑢 ↔ (0g𝑊)(lt‘𝑊)𝑣))
22 breq1 5061 . . . . . . . 8 (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
2322imbi2d 342 . . . . . . 7 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2423ralbidv 3197 . . . . . 6 (𝑢 = 𝑣 → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2521, 24anbi12d 630 . . . . 5 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))))
2625cbvrexvw 3451 . . . 4 (∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2720, 26sylib 219 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2819, 27r19.29a 3289 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
29 simpl1 1183 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
30 simpl3 1185 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
31 eqid 2821 . . 3 (+g𝑊) = (+g𝑊)
32 simpl2 1184 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → (oppg𝑊) ∈ oGrp)
33 simpr 485 . . . . . . . . . 10 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
34 ralnex 3236 . . . . . . . . . 10 (∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3533, 34sylibr 235 . . . . . . . . 9 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
36 rexanali 3265 . . . . . . . . . . . 12 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))
3736imbi2i 337 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
38 imnan 400 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3937, 38bitri 276 . . . . . . . . . 10 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4039ralbii 3165 . . . . . . . . 9 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4135, 40sylibr 235 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)))
4222notbid 319 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
4342anbi2d 628 . . . . . . . . . . 11 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4443rexbidv 3297 . . . . . . . . . 10 (𝑢 = 𝑣 → (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4521, 44imbi12d 346 . . . . . . . . 9 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥))))
4645cbvralvw 3450 . . . . . . . 8 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4741, 46sylib 219 . . . . . . 7 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4847r19.21bi 3208 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4914notbid 319 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
5013, 49anbi12d 630 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
5150cbvrexvw 3451 . . . . . 6 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
5248, 51syl6ib 252 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
53523impia 1109 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
54 simp1l1 1258 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ oGrp)
55 isogrp 30631 . . . . . . 7 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
5655simprbi 497 . . . . . 6 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57 omndtos 30634 . . . . . 6 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
5854, 56, 573syl 18 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ Toset)
59 simp2 1129 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑣 ∈ (Base‘𝑊))
601, 3, 4tltnle 30577 . . . . . . . . . 10 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
6160bicomd 224 . . . . . . . . 9 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
62613com23 1118 . . . . . . . 8 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
63623expa 1110 . . . . . . 7 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
6463anbi2d 628 . . . . . 6 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6564rexbidva 3296 . . . . 5 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6658, 59, 65syl2anc 584 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6753, 66mpbid 233 . . 3 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
681, 2, 3, 4, 5, 29, 30, 31, 32, 67archiabllem2 30754 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
6928, 68pm2.61dan 809 1 ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3138  wrex 3139   class class class wbr 5058  cfv 6349  Basecbs 16473  +gcplusg 16555  lecple 16562  0gc0g 16703  ltcplt 17541  Tosetctos 17633  Grpcgrp 18043  .gcmg 18164  oppgcoppg 18413  Abelcabl 18838  oMndcomnd 30626  oGrpcogrp 30627  Archicarchi 30734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-tpos 7883  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-seq 13360  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-ple 16575  df-0g 16705  df-proset 17528  df-poset 17546  df-plt 17558  df-toset 17634  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-grp 18046  df-minusg 18047  df-sbg 18048  df-mulg 18165  df-oppg 18414  df-cmn 18839  df-abl 18840  df-omnd 30628  df-ogrp 30629  df-inftm 30735  df-archi 30736
This theorem is referenced by: (None)
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